Chapter 10: Circles

Earlier classes treated circles as round figures with a radius and a centre, with formulas for circumference and area. This chapter adds a deeper layer: how lines interact with circles. A line drawn near a circle can miss it (no intersection), cut through it (a secant, two intersection points), or just touch it at exactly one point , a tangent.

The two key theorems of the chapter are remarkably clean: (1) the tangent at any point of a circle is perpendicular to the radius at the point of contact; (2) the lengths of two tangents drawn from an external point to a circle are equal. Almost every board problem on this chapter is a clever application of these two facts, often combined with the Pythagoras theorem or properties of triangles.

Tangent geometry shows up everywhere outside the textbook: the path of a wheel as it rolls on a road, the satellite footprint on the Earth's surface, optical reflections from curved mirrors, and the design of gears. The reasoning style , short, sharp deductions from one or two named theorems , is also a good warm-up for higher mathematics, where this style becomes the norm.

You will be expected to prove the two theorems and apply them to find lengths, angles, and to demonstrate concurrency or perpendicularity. Sketching neat diagrams is non-negotiable here; about half the marks on a typical board problem are earned by the correctly labelled figure.

What's inside

• Tangent vs secant , definitions and existence of tangents.
• Theorem 1 , the radius is perpendicular to the tangent at the point of contact.
• Theorem 2 , two tangents from an external point have equal length.
• Quadrilaterals circumscribing a circle , sum of opposite sides equal.
• Computation problems , lengths, angles, and standard configurations.

Key results / Formula card

• A tangent to a circle is a line that meets the circle in exactly one point.
• Theorem (tangent-radius): If $\ell$ is a tangent at point $P$ to a circle of centre $O$, then $OP \perp \ell$.
• Theorem (equal tangents): From an external point $T$, the two tangent lines to the circle touch the circle at points $A$ and $B$. Then $TA = TB$.
• Corollary: $\triangle OAT \cong \triangle OBT$ (RHS), so the line $OT$ bisects $\angle ATB$ and $\angle AOB$.
• Tangent length: For an external point at distance $d$ from the centre, the tangent length is $\sqrt{d^2 - r^2}$.
• Circumscribed quadrilateral: If $ABCD$ has all four sides tangent to a circle, then $AB + CD = BC + DA$.